Question

# 2: Consider the real symmetric matrix A= 4 1 a) What are the eigenvalues and eigenvectors. [Hint: Use wolframalpha.] b) What is the trace of A, what is the sum of the eigenvalues of A. What is a general theorem th c) The eigenvalues of A are real. What is a general theorem which assert conditions that t d) Check that the eigenvectors are real. What is a general theorem which asserts conditions th asserts equality? eigenvalues are real and applies to this case. the eigenvectors can be taken to be real ? e) Here the eigenvalues, say X1, 2 are different, distinct. Let ci,c2 be eigenvectors for these wri as columns Form the 2 by 2 matrix B (c1,2) with columns c1,c2 and the diagonal matrix Compute directly X=B-D-B-1 What is the relation of X with the original matrix A? We say A is diagonalizable in this case. f) Using the above compute the matrix Al00 directly. Hint: What is the relation to B. D100. g) What is a general theorem about square matrices X, that applies to A above, that ensures ? If related compute the last.] we may form B and D and relate B.D.B-1 to X? That is, what is a general theorem al diagonalizability? h) The characteristic polynomial of A above is det -4 λ-1 Show that it is 2-2 t 15 You will recognise 2-trace of A,-15 = det (A). Show directly that M-2A-151d = 0 with ld = ( , What is the general theorem that asserts this fact that applies to A?

Answer 1

a) [1 - lambda 4 4 1 - lambda] = 0, Rightarrow (1 - lambda)^2 - 16 = 0, Rightarrow (1 - lambda + 4)(1 - lambda - 4) = 0 Rigtarrow (lambda - 5) (lambda + 5) = 0 Rightarrow lambda = -3, 5 Therefore the eigen values are -3, 5 b) Trace A = 1 + 1 = 2, sum of the eigen values of A = -3 + 5 = 2 theorem: The trace of a matrix is the sum of it's eigen values c) Here the eigen values are real. Theorem: A matrix has all real eigen values and orthonormal real eigen vectors iff it is real symmetric. d) Let X = (x_1 x_2) be an eigen vector corresponding to -3 Then AX = -3X gives x_1 + 4x_2 = -3x_1 4x_1 + x_2 = -3x_2 i.e, x_1 + x_2 = 0 We take x_2